PISEMA powder patterns and PISA wheels

Resonance patterns observed in 2D PISEMA (polarization inversion spin exchange at magic angle) spectra from a transmembrane alpha-helix have been demonstrated to yield structural details of the protein. This paper presents a mathematical discussion of the PISEMA powder spectrum as the image in the frequency plane of a quadratic function from the sphere of unit vectors. The simplicity of this function allows easy calculation of the powder spectrum. Based on this analysis of powder patterns, four degeneracies are discussed which arise in determining possible orientations associated with PISA spectra. This paper also gives parametric equations for PISA wheels, which are specific patterns observed in PISEMA spectra of oriented peptides. These wheels are useful both in assigning the resonances and in determining the orientation of the helix with respect to the magnetic field. The union of these PISA wheels gives the entire powder spectrum.